Fracture and debonding of a thin film on a stiff substrate: analytical and numerical solutions of a one-dimensional variational model

We study multi-fissuration and debonding phenomena of a thin film bonded to a stiff substrate using the variational approach to fracture mechanics. We consider a reduced one-dimensional membrane model where the loading is introduced through uniform inelastic (e.g., thermal) strains in the film or imposed displacements of the substrate. Fracture phenomena are accounted for by adopting a Griffith model for debonding and transverse fracture. On the basis of energy minimization arguments, we recover the key qualitative properties of the experimental evidences, like the periodicity of transverse cracks and the peripheral debonding of each regular segment. Phase diagrams relate the maximum number of transverse cracks that may be created before debonding takes place, as a function of the material properties and the sample’s geometry. The theoretical results are illustrated with numerical simulations obtained through a finite element discretization and a regularized variational formulation of the Ambrosio–Tortorelli type, which is suited to further extensions in two-dimensional settings.